When the two thin lenses are put in contact then which of the following is the focal length of the combination?

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The geometric mean of the two lengths

The same as the larger focal length.

Greater than either focal length

Smaller than either focal length

answer is D.

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Detailed Solution

When the two thin lenses are put in contact, the focal length of the combination is smaller
than either focal length.
Thin Lens:
A thin lens refers to a lens with a thickness that is negligible compared to its radii of
curvature.
A lens's power is determined by multiplying its focal length by itself (in metres). When one occurs when two lenses are combined and in contact, the combined effective power is the sum of the unique abilities. The lens combination's combined focal length

(F_{c})

is determined by the formula

\frac{1}{F_{c}} = \frac{1}{F_{1}} + \frac{1}{F_{2}} + \frac{1}{F_{3}} + . . . . . . . . . . + \frac{1}{F_{n}}

Where

F_{c}

is the combined focal length,

F_{1}

is the focal length of the first lens,

F_{2}

is the focal
length of the second lens and

F_{n}

is the first focal length of the nth lens placed in series,
Case 1:

F_{1}

F_{2}

F_{2}

F_{1}

Combined focal length:

\frac{1}{F_{c}} = \frac{1}{F_{1}} + \frac{1}{F_{2}}

Now

F_{1} = F_{2} = F

\Rightarrow F_{c} = \frac{F_{1} \times F_{2}}{{F_{1} + F_{2}}_{}}

\Rightarrow F_{c}

\frac{F_{} \times F_{}}{{F_{} + F_{}}_{}}

\Rightarrow F_{c}

\frac{F^{2}}{2 F}

\Rightarrow F_{c}

\frac{F}{2}

\Rightarrow F_{c} = 0.5 f

So,

F_{c}

< F
The combined focal length is lesser than both focal lengths.
Hence when the two lenses are put close to each other, the combined focal length is lesser
than the smaller focal length.

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